# Estimation in Semiparametric Models

## Some Recent Developments

• Johann Pfanzagl
Book

Part of the Lecture Notes in Statistics book series (LNS, volume 63)

1. Front Matter
Pages i-iii
2. ### Introduction

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Pages 1-1
3. ### Survey of basic theory

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Pages 2-3
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Pages 4-6
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Pages 7-16
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Pages 17-22
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Pages 23-34
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Pages 35-37
4. ### Semiparametric families admitting a sufficient statistic

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Pages 38-47
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Pages 48-52
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Pages 53-87
5. ### Auxiliary results

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Pages 88-105
6. Back Matter
Pages 106-112

### Introduction

Assume one has to estimate the mean J x P( dx) (or the median of P, or any other functional t;;(P)) on the basis ofi.i.d. observations from P. Ifnothing is known about P, then the sample mean is certainly the best estimator one can think of. If P is known to be the member of a certain parametric family, say {Po: {) E e}, one can usually do better by estimating {) first, say by {)(n)(.~.), and using J XPo(n)(;r.) (dx) as an estimate for J xPo(dx). There is an "intermediate" range, where we know something about the unknown probability measure P, but less than parametric theory takes for granted. Practical problems have always led statisticians to invent estimators for such intermediate models, but it usually remained open whether these estimators are nearly optimal or not. There was one exception: The case of "adaptivity", where a "nonparametric" estimate exists which is asymptotically optimal for any parametric submodel. The standard (and for a long time only) example of such a fortunate situation was the estimation of the center of symmetry for a distribution of unknown shape.

### Keywords

DEX boundary element method development distribution eXist estimator function functional measure median probability probability measure symmetry time

#### Authors and affiliations

• Johann Pfanzagl
• 1
1. 1.Mathematisches InstitutUniversität zu KölnKöln 41Federal Republic of Germany

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-3396-1
• Copyright Information Springer-Verlag New York 1990
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-97238-1
• Online ISBN 978-1-4612-3396-1
• Series Print ISSN 0930-0325
• Buy this book on publisher's site